Differential Geometry of Singular Spaces

奇异空间的微分几何

• 批准号: 5407091
• 负责人: Professor Dr. Andreas Bernig
• 金额: --
• 依托单位: Schwerpunkt Analysis und Numerik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2003
• 资助国家: 德国
• 起止时间: 2002-12-31 至 2005-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5407091?language=en
• 关键词: Differential; Geometry; Singular; Spaces

The study of Riemannian manifolds is central in mathematics. They are smooth objects which can be analyzed using geometric and analytic tools. It turned out over the last fifteen years that the study of certain limits of Riemannian manifolds is very fruitful. In general, these limits present difficult singularities. Examples for limit spaces are metric spaces with a lower or an upper curvature bound. In this project, we want to use Geometric Measure Theory in order to construct and study other types of limit spaces which are better suited for questions about scalar curvature and Ricci curvature. There will be an interesting interplay between Differential Geometry, Geometric Measure Theory, Convex Geometry and Subanalytic Geometry. Hopefully, there will be applications of this theory to the existence of Einstein metrics, realization of Yamable invariants and to the geometry of Riemannian manifolds with scalar or Ricci curvature bounds.

Riemannian歧管的研究在数学中至关重要。它们是可以使用几何和分析工具来分析的光滑对象。事实证明，在过去的十五年中，对Riemannian歧管的某些限制的研究非常富有成果。通常，这些限制具有困难的奇异性。极限空间的示例是具有较低或上部曲率结合的度量空间。在这个项目中，我们希望使用几何测量理论，以构建和研究其他类型的极限空间，这些空间更适合有关标量曲率和RICCI曲率的问题。差异几何形状，几何测量理论，凸几何和亚分析几何形状之间将有一个有趣的相互作用。希望该理论将在爱因斯坦指标的存在，俗称不变式的实现以及带有标量或RICCI曲率边界的Riemannian歧管的几何形状上应用。